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Mathematics > Category Theory

Title:
Algebraic weighted colimits

Abstract: In this thesis weighted colimits in 2-categories equipped with promorphisms
are studied. Such colimits include most universal constructions with counits,
like ordinary colimits in categories, weighted colimits in enriched categories,
and left Kan extensions.
In the first chapter we recall the notion of 2-categories equipped with
promorphisms (also simply called equipments), that provide a coherent way of
adding bimodule-like morphisms to a 2-category. In the second chapter we recall
two ways of defining weighted colimits in equipments. Most important to us is
their original definition, introduced by Wood, in equipments that are endowed
with a closed structure. The second notion, of what we call pointwise weighted
colimits, was introduced by Grandis and Par\'e. It requires no extra structure
and generalises Street's notion of pointwise left Kan extensions in
2-categories. The main result of the second chapter gives a condition, on
closed equipments, under which these two notions coincide.
In the third chapter we consider monads on equipments. The main idea of this
thesis, given in the fourth chapter, generalises the notions of lax and colax
morphisms, of algebras over a 2-monad, to notions of lax and colax
promorphisms, of algebras over a monad on an equipment. One of these, that of
right colax promorphisms, is well suited to the construction of weighted
colimits. In particular, given a monad T on an equipment K, we will show that
T-algebras, colax T-morphisms and right colax T-promorphisms form a double
category T-rcProm. Although weaker than equipments, double categories still
allow definition of weighted colimits, and our main result states that the
forgetful functor T-rcProm -> K lifts all weighted colimits whenever K is
closed, under some mild conditions on T.